Briefly, we consider the problem when a tangent line to a curve has zero slope. That is, the derivative of the curve is zero at some point.
Such a point is called a “critical point” and tells us about extreme values.
Example. Consider the function . We plot this out:
Now, we can consider the tangent line at, say, . We draw this tangent line in red:
Observe every point on the line has the property that . In other words, is a “Global Minimum” (it’s less than every other if ).
How can we detect such “extrema” (i.e., “maxima” and “minima”)? Well, the tangent line has zero slope. Equivalently: when the point is an extreme point.
Sometimes these “extrema” points are local. For example, plotting when :
So how do we detect extrema?
Well, we take our function , take its derivative, then find points such that .
This is the general routine, but lets leave the motivating post with a few questions…
Problem: as we can see with the function , some extrema are local while others are global. How do we
(a) Determine if an extrema is a maximum or minimum?
(b) Determine if the extrema is local or global?